Propagation and attenuation characteristics of free flexural waves in multi-stepped periodic beams by the method of reverberation-ray matrix

Abstract

This paper aims to present better physical understanding of flexural wavenumber spectral behaviors of multi-stepped periodic beams by the method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory. According to the Bloch theorem, the formulation of MRRM is presented and the complex flexural wavenumbers of multi-stepped periodic beams are solved by numerical techniques. The validation of MRRM in analyzing flexural wave behaviors of multi-stepped periodic beams is verified by comparison with the analytical wavenumber spectrum of a homogeneous beam. Numerical examples are provided to illustrate the general wavenumber spectral characteristics of an infinite periodic beam with alternate variation of cross-sections. The effects of the material properties and geometrical parameters of the beam segments are evaluated. The most remarkable finding inspired from the wavenumber spectral curves is that the real wavenumber of the multi-stepped periodic beam does not vary with frequency in the attenuation bands, which indicates that the wavelength of the flexural wave is constant in the attenuation bands.

Highlights

• A multi-stepped periodic beam model with arbitrary finite number of beam segments is established.

• Complex wavenumber spectrum analysis is performed to reveal the propagation and attenuation behaviors of multi-stepped periodic beams.

• Similarity and comparative studies on the propagation and attenuation characteristics of beams of a uniform cross-section and of variable cross-sections are carried out to provide better physical understanding.

• An innovative finding that the wavelength of the flexural wave is constant in the attenuation bands is achieved.

GRAPHICAL ABSTRACT

Keywords:

• Propagation and attenuation characteristics
• free flexural wave
• multi-stepped periodic beam
• complex wavenumber spectral curves
• the method of reverberation-ray matrix

Acknowledgments

The authors are grateful for the financial support. The authors would also like to sincerely thank the anonymous reviewers for the critical and constructive comments on this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is financially supported by National Natural Science Foundation of China [grant number 52001113], Fundamental Research Funds for the Central Universities [grant number B200202059], Natural Science Foundation of Jiangsu Province [grant numbers BK20190485 and BK20190967], Natural Science Foundation for Colleges and Universities in Jiangsu Province [grant number 19KJB580010].